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# 数学代写|数值分析代写Numerical analysis代考|MATH408 ERRORS: DEFINITIONS, SOURCES, AND EXAMPLES

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## 数学代写|数值分析代写Numerical analysis代考|ERRORS: DEFINITIONS, SOURCES, AND EXAMPLES

The error in a computed quantity is defined as
Error $=$ true value $-$ approximate value
The relative error is a measure of the error in relation to the size of the true value being sought:
$$\text { Relative error }=\frac{\text { error }}{\text { true value }}$$
To simplify the notation when working with these quantities, we will usually denote the true and approximate values of a quantity $x$ by $x_T$ and $x_A$, respectively. Then we write
\begin{aligned} \text { Error }\left(x_A\right) &=x_T-x_A \ \operatorname{Rel}\left(x_A\right) &=\frac{x_T-x_A}{x_T} \end{aligned}
As an illustration, consider the well-known approximation
$$\pi \doteq \frac{22}{7}$$
Here $x_T=\pi=3.14159265 \ldots$ and $x_A=22 / 7=3.1428571, \ldots$,
Error $\left(\frac{22}{7}\right)=\pi-\frac{22}{7} \doteq-0.00126$
$\operatorname{Rel}\left(\frac{22}{7}\right)=\frac{\pi-(22 / 7)}{\pi} \doteq-0.000402$
Another example of error measurement is given by the Taylor remainder (1.9).
An idea related to relative error is that of significant digits. For a number $x_A$, the number of its leading digits that are correct relative to the corresponding digits in the true value $x_T$ is called the number of significant digits in $x_A$. For a more precise definition, assuming the numbers are written in decimal, calculate the magnitude of the error. $\left|x_T-x_A\right|$. If this error is less than or equal to five units in the $(m+1) s t$ digit of $x_T$, counting rightward from the first nonzero digit, then we say $x_A$ has, at least, $m$ significant digits of accuracy relative to $x_T$.

## 数学代写|数值分析代写Numerical analysis代考|Sources of Error

Imagine solving a scientific-mathematical problem, and suppose this involves a computational procedure. Errors will usually be involved in this process, often of several different kinds. We will give a rough classification of the kinds of error that might occur.
(E1) Mathematical equations are used to represent physical reality, a process that is called mathematical modeling. This modeling introduces error into the realworld problem that you are trying to solve. For example, the simplest model for population growth is given by
$$N(t)=N_0 e^{k t}$$
where $N(t)$ equals the population at time $t$, and $N_0$ and $k$ are positive constants. For some stages of growth of a population, when it has unlimited resources, this can be an accurate model. But more often, it will overestimate the actual population for large $t$.

The error in a mathematical model falls outside the scope of numerical analysis, but it is still an error with respect to the solution of the overall scientific problem of interest.
(E2) Blunders and mistakes are a second source of error, a familiar one to almost everyone. In the precomputer era, blunders generally consisted of isolated arithmetic errors, and elaborate check schemes were used to detect them. Today the mistakes are more likely to be programming errors. To detect these errors, it is important to have some way of checking the accuracy of the program output. When first running the program, use cases for which you know the correct answer. With a complex program, break it into smaller subprograms, each of which can be tested separately. And when you believe the entire program is correct and are running it for cases of interest, maintain a watchful eye as to whether the output is reasonable.
(E3) Many problems involve physical data, and these data contain observational error. For example, the speed of light in a vacuum is
$$c=(2.997925+\epsilon) \cdot 10^{10} \mathrm{~cm} / \mathrm{sec}, \quad|\epsilon| \leq 0.000003$$

## 数学代写|数值分析代写数值分析代考|ERRORS: DEFINITIONS, SOURCES, AND EXAMPLES

error $=$ 真值 $-$ 相对误差是相对于正在寻找的真实值大小的误差的度量:
$$\text { Relative error }=\frac{\text { error }}{\text { true value }}$$在处理这些量时，为了简化符号，我们通常表示一个量的真值和近似值 $x$ by $x_T$ 和 $x_A$，分别。然后我们写
\begin{aligned} \text { Error }\left(x_A\right) &=x_T-x_A \ \operatorname{Rel}\left(x_A\right) &=\frac{x_T-x_A}{x_T} \end{aligned}作为一个例子，考虑众所周知的近似$$\pi \doteq \frac{22}{7}$$

$\operatorname{Rel}\left(\frac{22}{7}\right)=\frac{\pi-(22 / 7)}{\pi} \doteq-0.000402$另一个误差测量的例子是泰勒余数(1.9)。

## 数学代写|数值分析代写Numerical analysis代考|Sources of Error

$$N(t)=N_0 e^{k t}$$

$$c=(2.997925+\epsilon) \cdot 10^{10} \mathrm{~cm} / \mathrm{sec}, \quad|\epsilon| \leq 0.000003$$

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